A006630 From generalized Catalan numbers.

1, 6, 33, 182, 1020, 5814, 33649, 197340, 1170585, 7012200, 42364476, 257854776, 1579730984, 9734161206, 60290077905, 375138262520, 2343880406595, 14699630061270, 92502956574105, 583920410197950, 3696470074992240, 23461536762704040, 149270218961671548

Offset: 0

Author

Simon Plouffe

Keywords

• nonn
• easy
• changed

Comments

It appears that this is the self-convolution of A001764 starting 1, 3, 12, ... . - Alon Regev, Aug 07 2015

References

• H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
• Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

Crossrefs

Column 3 of A092276.

Closely related to A000139.

Cf. A001764, A006631.

Programs

Magma

[2*Binomial(3*n+6, n)/(n+2): n in [0..25]]; // Vincenzo Librandi, Aug 07 2015

Mathematica

Table[2 Binomial[3 n+6,n]/(n+2), {n, 0, 25}] (* Vincenzo Librandi, Aug 07 2015 *)
CoefficientList[Series[(-1 + (2*Sin[(1/3)*ArcSin[(3*Sqrt[3]*Sqrt[x])/2]]) / (Sqrt[3]*Sqrt[x]))^2/x^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 03 2022, after Vladimir Kruchinin *)

Maxima

taylor(((1/sqrt(3/4*x)*sin(1/3*asin(sqrt(27/4*x)))-1)/x)^2,x,0,17); /* Vladimir Kruchinin, Oct 03 2022 */

Maxima

makelist(2*binomial(3*n+6, n)/(n+2),n,0,30); /* Vladimir Kruchinin, Oct 03 2022 */

PARI

a(n) = 2*binomial(3*n+6, n)/(n+2); \\ Andrew Howroyd, Nov 06 2017

SageMath

def A006630(n): return 2*binomial(3*(n+2),n)//(n+2)
print([A006630(n) for n in range(41)]) # G. C. Greubel, Aug 31 2025

Formula

G.f.: hypergeometric3_F_2([ 2, 8/3, 7/3 ], [ 4, 7/2 ], 27*x/4).

a(n) = 2*binomial(3*n+6, n)/(n+2). - Henry Bottomley, Sep 24 2001

G.f.: (1 - RootOf(x-t*(1-t)^2,t))^(-6) (algebraic function in Maple notation). - Mark van Hoeij, Nov 08 2011

G.f.: ((1/sqrt((3/4)*x)*sin((1/3)*asin(sqrt((27/4)*x)))-1)/x)^2. - Vladimir Kruchinin, Oct 03 2022

a(n) = (n+1)/2 * A000139(n+2). - F. Chapoton, Feb 23 2024

Extensions

More terms from Christopher Lund (clund(AT)san.rr.com), Apr 16 2002

a(21)-a(22) from Vincenzo Librandi, Aug 07 2015